Question

The equation of tangent to the curve $y = \cos(x + y)$ where $-2\pi \le x \le 2\pi$ and which is parallel to the line $x + 2y = 0$, is

• A. $2x + 4y + \pi = 0$
• B. $2x + 4y - \pi = 0$
• C. $2x + 4y - 3\pi = 0$
• D. $2x - 4y + 3\pi = 0$

Hint:

For curves involving trigonometric functions, check if the calculated point satisfies the original curve equation.

Answer 1

The Correct Option is B

Step 1: Concept
The slope of the tangent $dy/dx$ must equal the slope of the parallel line $(-1/2)$.

Step 2: Meaning
Differentiating $y = \cos(x + y)$: $dy/dx = -\sin(x + y)(1 + dy/dx)$.
$-1/2 = -\sin(x + y)(1 - 1/2) \implies -1/2 = -\sin(x+y)(1/2) \implies \sin(x+y) = 1$.

Step 3: Analysis
If $\sin(x+y) = 1$, then $\cos(x+y) = 0$. Since $y = \cos(x+y)$, we have $y = 0$.
$\sin(x+0) = 1 \implies x = \pi/2$ (within range). Point is $(\pi/2, 0)$.

Step 4: Conclusion
Equation: $y - 0 = -1/2(x - \pi/2) \implies 2y = -x + \pi/2 \implies 2x + 4y - \pi = 0$. Final Answer: (B)